Particles have properties that seemed tied to some internal
structure. This is similar to the color of a ball. The property is an inherent
feature of the ball.† Its motion
(momentum) is tied to its space-time properties.† Internal properties are independent of space
time

Intrinsic properties standard

charge, mass† (Usually
we choose eigenstates of definite mass and charge) [Mass referred to here is the
rest mass of the particle.]

Spin seems to bridge the gap. There is some internal
structure associated with particles that provides the particle with an
intrinsic angular momentum.† This behaves
in the standard way that angular momentum behaves and must be added to orbital
angular momentum. An electron can have its spin reoriented from right to left
pointing.† This constitutes a change of
angular momentum of 1 unit [].† If this were part
of the change associated with an atomic transition then the electromagnetic
field would need to have the compensating angular momentum change to keep
angular momentum conserved.

Rotations are our go to transformations:

†There are unitary
operators that rotate quantum states.

There are unitary operators that translate states.

Unitary is the complex space analog of orthogonal. Rotations
do not change the length of a vector. Unitary transformations do not change the
ďamountĒ of quantum states. This can be more formally stated in terms of
probability.

Note: If you choose the standard position space
representation for the momentum operator †where for simplicity
we consider only one dimension and if we remember the Taylor
expansion as the tool to interpret. One can show that the translation operator will move a
function a distance.